Have you wondered why, if you put something on a slope, it rolls or slides down? Gravity is a factor, but does it explain the complete picture? In this article, let’s explore the properties of slopes (inclined planes) and why objects move there.
The Dynamics of the Inclined Plane
First, which forces are acting on an object on an inclined plane? Obviously, there’s the gravitational force, but there’s also another force, called the normal force, acting on it. When an object is in contact with a surface, the surface exerts a supporting force perpendicular to the surface of contact. For example, if you put a box on your table, neither the box nor the table moves. As the box exerts its weight onto the table, the table exerts a normal force to support it and keep it where it is.
But since the plane is inclined, the normal force is not pointing vertically upward. Remember that if multiple forces act on the same object, its motion will be the combined effect of these forces. Therefore, as the inclined normal force and the gravitational force act together, the object moves in a direction that brings it down the slope.
Note: the effects of friction is not included in this diagram
However, if you put a ball on a slope, it rolls down instead of skidding down. Why does that happen? As the ball moves, the friction and the gravitational force act together and exert torque on the ball. This causes it to spin, and when combined with movement, it rolls.
The blue arrows represent forces that cause the ball to rotate
How to Calculate These Forces?
After learning about the general principle, you might wonder how these forces are calculated. First, remember that the weight of an object (which is the downward gravitational force on an object) is mg, where m is the mass and g is the gravitational acceleration. The normal force applies equally and oppositely to that object’s weight, so the force is also mg. But for inclined planes, it’s different.
Imagine that you have a box on a surface that is perpendicular to the ground. Could it exert any normal force? The obvious answer is no, because the box falls straight down (neglecting the effects of friction). Therefore the normal force is zero. If the resulting answer is 1 (the entire mass) at 0 degrees and 0 at 90 degrees, which common function will you think of? The answer is the cosine function, which is also the correct function for this purpose. The normal force exerted on an object on a slope with an angle θ is cos(θ) * mg. If you don’t understand what this means, refer to the below illustration of the force vectors:
Finally, you add these vectors together, and you get a net force of sin(θ) * mg pushing the object down the slope, without including the friction. The direction of the net vector is equal to that of the direction of the inclined plane (pointing down), so no other forces are pushing the object in other directions in an ideal case.
Note that m equals the mass of the object and g is the gravitational acceleration, such that mg is the weight of the object
Conclusion
In this article, we explored the dynamics of an inclined plane (a slope) and explained why objects slide or roll down these slopes without outside intervention. It comes down to the effects of the gravitational force and the normal force, which is not pointing straight up. If you would like more content included, please leave your suggestions in the comments below.